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The Information in the Term Structure of German Interest Rates

Gianna Roero’), Foruhar Madjlessi*) and Costanza Torricelli’~

This paper investigates the informative content of the term structure for German interest rates. Most of the empirical work on this subject has concentrated on tests of the Expectations Hypothesis by using US interest rate data, and the hypothesis has been found to be rejected in the great majority of cases. However, term structure information failures may simply be a reflection of the specific data used and of the particular framework within which the hypothesis has been tested. In fact, in most empirical work the information in the term structure is represented only by some measure of its slope, while an important element, that is a time- varying term premium, is missed from the specification of the models. This paper examines further the information content of the term structure using monthly data for Germany and, following the general equilibrium theory of the term premium, extends the standard framework of the EH by including a measure of the riskless rate volatility in the information set. Our results indicate that the German term structure appears to forecast future short term interest rates surprisingly well, compared with previous studies with US data. Moreover, and in line with previous findings, the slope of the term structure alone does not have predictive power for long term interest rate movements, though the direction suggested by the coefficient estimates is correct. Finally, inclusion of a volatility term represents an improvement in most regressions for the short rate, and significantly improves those for the long rate.

Cet article ttudie le contenu informatif de la structure P terme des taux d’int&et Allemande. La plupart des recherches empiriques sur cet argument s ‘est concent&. sur des tests de 1’Hypothbse des Expectatives en utilisant des don&es sur les taux d ‘in&$ pour les Etats Unis, et 1 ‘Hypothbse B et6 refusde dans la plupart des cas. Toutefois, les insuc&s des tests sur le contenu informatif des taux d’interet pourraient dependre des don&es specitiques utilis6es et du contexte particulier ou l’hypothese a et6 test&. En effect, dans la plupart des travaux empiriques, 1 ‘information dans la structure a terme est repr&ent& seulement par une mesure de son inclinaison, tandis que la specification de ces modeles n’a pas un Clement important, c ‘est a dire le prix a risque variable.

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Cet article examine aussie le contenu informatif de la structure h terme en utilisant des donnk mensuelles pour 1 ‘Allemagne, et en suivant la thhie de 1 ‘kquilibre ghkal du prix h risque, cet article-ci dkveloppe le contexte standard de la thhie des expectatives en incluent une mesure de la volatiliti du taux d ‘in&et h bref d&i dans le set d’information. Nos r&hats indiquent que la structure h terme Allemande prhoit les taux d ‘M&et h bref d&i d ‘une manihe surprenante, face a les etudes ptientes sur les taux d ‘in&et des Etats Unis. L ‘inclinaison de la structure a terme toute seule n ‘a pas du pouvoir de prkvision pour les taux d ‘in&et h long dBai, meme si la direction sugg&tk par les estimations des coefficients est correcte. En concluant, 1 ‘inclusion de la volatilid reprksente une amklioration dans la plupart des r6gressions pour les taux d ‘in&et h bref d&i, et amkliore d ‘une man&e significative les r6gressions pour les taux d’intket a long d&i.

Information, interest rate, term structure, term premia, volatility.

University of Cagliari and CRENoS, Viale Fra’ Igmzio, 78, 09123 Cagliari (Italy); Tel: + 39-70-6753763, Fax: + 39-70-6753160, E-mail: boem@vaxcaI.unica.it University of Karlsruhe, Postfach 6980, 76128 Karlsruhe (Germany); Tel: + 49-721- 6083427, Fax: + 49-721-359200, E-mail: madjlessi~.wiwi.uai-karlsruhe.de University of Modena, Via Berengario, 51, 41100 Modem (ItaIy); Tel: + 39-59-417733, Fax: + 39-59-417937, E-mail: torricelli@unimo.it

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1. INTRODUCTION’

Most recent stochastic models of the term structuren do not deal with an issue that

has been long debated within the Expectations Theory, i.e. the informative content of

the term structure of interest rates.

The relationship between short and long rates has macroeconomic implications in that

it is central to monetary transmission mechanism and hence to macroeconomic policy.

In fact, while the monetary authority can mainly control short-term rates, the aggregate

demand depends essentially on long-term rates, hence the term structure of interest rates

plays an important role in the monetary transmission mechanism.

Therefore the analysis of the information content of the term structure establishes a

connection between modern theory of finance and economic policy.

Purpose of the present paper is to take a fresh look at the issue in order to discuss

and test whether results stemming from stochastic models of the term structure can

contribute to a better understanding of the proposed information issue.

A great deal of empirical research on the term structure of interest rates has focused

upon the Expectations Hypothesis (EH) that essentially relates long-term yields to actual

and expected future short rates. This paper examines whether the slope of the term

structure can alone explain titure movements on the short rates or whether these can be

better explained with the inclusion of a time-varying risk premium. The issue is

explored by means of standard regression analysis with monthly data derived from the

estimation of the German term structure over the period 1983( 12) - 1994( 12).

The paper is organised as follows, In section 2 we review the standard regression

tests of the EH and in section 3 we examine the role of time-varying term premia and

stochastic models. In section 4 we present the data and in section 5 we analyse their

properties, In section 6 we use the regression analysis to test the information content of

the German term structure, and in section 7 we extend the standard framework by

including a measure of the riskless rate volatility in the information set. Section 8

concludes. Notation and definitions are given in the Appendix.

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2. THE INFORMATION ISSUE .

The informationm in the Term Structure (TS) has been empirically analysed by

means of linear regression equations: the existing models essentially differ in the choice

of the regressors and of the regressand. The former is represented by some measure of

the “slope” of term structure, the latter by the information required.

The point is now: what gives information on what? According to the Expectations

Hypothesis (EH) framework, long rates are averages of current and expected future

short rates. This implies that an upward (downward) sloping TS curve predicts an

increase (decrease) in short rates.“’

The literature has investigated the informational content of the term structure using

different spreads as measures of the term structure thus obtaining different pieces of

information, i.e. using different regression equations.

The specification of a linear regression equation consists of the identification of the

regressor(s) and of the regressand, which means, in the present context, the source(s) of

information and the information obtainable respectively. Plausible regressors are various

type of term structure spreads (forward-spot rate differential, long-short rate

differential), regressand of interest are: term premia, changes in the spot rate, changes in

inflation and changes in the real return. In other words, the term structure of interest

rates can be tested both as a predictor of future rateseand as a predictor of future

inflation. In the present paper only the former case is considered.

2.1 The term structure as a predictor of future interest rates

The current term structure can provide information about the future term structure:

the precise content of this statement has been evaluated in a series of papers by Eugene

Fama and others, Among them we recall only those that are most directly related to the

present discussion: Fama (1984,1990), Fama and Bliss (1987) and Jorion and Mishkin

(1990).”

The papers are essentially based on estimation of the one or all the following

regressions:

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(l)r(T- 1.T) - r(t,t +l) = a +P[f(t,T- l,T) - r(t,t +1)]+&(T)

(2) h(t,t + l,T) - r(t,t + 1) = a’+Bff(t,T - 1,T) - r(t,t + I)] + a’(t + 1)

(3)r(T- l,T)- r(t,t +l) = cl+E[r(t,T) - r(t,t + l)]+@(T)

(4) h(t,t +l,T) - r(t,t + 1) = a + i[r(t,T) - r(t,t +l)]+ i (t +l)

where 1 is the time unit and 2% and the rates are defined in the Appendix.

Equation (3) and (4) have been extended in Fama (1990) so to include the spot rate

level r(t, t+ I) as a regressor.

Fama (1984) estimates regressions (1) and (2) using averages of end-of-month bid

and asked prices for one- to six-month U.S. Treasury bills for the 1959-82 period. He

finds that while the forward-spot differential has always power as a predictor of future

holding term premium, the same is not true as for the future change in the one-month

spot rate. In the latter case, the author shows that variation through time of the expected

holding term premia contained in the forward ratevi obscures the power of the forward

rate as a predictor of future spot rates. By means of additional regressions, the author

shows that both the expected premia and the expected future spot rate components of

the forward rate are time-varying.

The latter assertion is also at the basis of Fama and Bliss (1987) paper, which extends

the previous analysis using longer maturity U.S. Treasury bills data, i.e. annual Treasury

maturities to five years for the period 1964-85. Main findings are: (i) one-year expected

returns for maturities up to 5 years net of the interest rate on a one year bond are time-

varying and seem to be directly related to business cycle; (ii) forward rate cannot

forecast near-term changes in the spot rate but still have forecast power for changes in

the one-year spot rate 2 to 4 years ahead; such a property reflects a mean-reverting

tendency of interest rates which is more evident at longer horizons.vii

Jorion and Mishkin (1991) also estimate regression (2) but using an international data

set made of monthly observations of government bonds of maturity from 1 up to 5

years, from August 1973 to June 1989 for the U.S., Britain, Germany and Switzerland.

Unlike Fama and Bliss (1987) they find that “the forward-spot spread has no significant

power to forecast changes in U.S. interest rates [...I. One possible explanation for the

difference is the shorter sample period used here[...].“. The results for the other

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countries are not more encouraging and quite consistent across countries, with the

exception of Germany and Switzerland that have a significant regression coefficient at

the longest 5 years term.

In Fama (1990) the focus shifts on the predictive power of the yield spread, i.e. the

difference between the long and the short rate. Since yield spreads for different

maturities are highly correlated, the author takes as a common spread the five-year

spread and estimates regressions of the type described by equations (3) and (4) using

end-of-month U.S. Treasury bond prices from June 1952 to December 1988 for

discount bonds with annual maturities from one to five years. The yield-spread can

always forecast the expected term premium, which turns out to increase with maturity

and to vary through time as emerged in the previous studies. Yet, the yield spread has

no power to forecast the one-year spot rate one year ahead and, once again as in earlier

papers, its predictive power increases with the forecast horizon. As reported in the next

section, Fama explains the latter finding decomposing the nominal rate into a real and an

inflation component.

Hence, evidence on the forecast power of the yield spread shows big similarities with

evidence on the forecast power of the forward-spot spread. In particular, the time-

varying nature of the term premium both in the yield- and in the forward-spot spread

seems to obscure the information about expected changes in the spot rate at the near

horizon,

Summing up: the term structure (represented either by a forward-spot spread or a

yield spread) is a good predictor of expected holding period term premia both at near

and at long horizons and a good predictor of expected spot rates only at longer

horizons. Time varying term premia play an important role in explaining the poor

performance of regressions (1) and (3) over near term horizons. The issue is central to

this paper and will be extensively discussed in the next section.

3. THE INFORMATION MISSED AND TIME-VARYING TERM PREMIA

The failures emerging from the tests on the informative content of the term

structure have been explained in the literature in many alternative ways. Among them

(for a discussion of alternative explanation see Torricelli( 1995)) the existence of time-

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varying term premia seems to be the most appealing explanation. Yet, time-varying

term premia cannot be easily reconciled with the EH. The point is clearly made by

Mankiw and Summers (1984): “Note that once it is extended to include a time-varying

liquidity premium, the expectations theory becomes almost vacuous. The liquidity

premium is a deus ex machina. Without an explicit theory of why there is such a

premium and why it varies, it has no tinction but tautologically to rescue the

theoty.[...] These results suggest the importance of developing models capable of

explaining fluctuating liquidity premiums.[, ,] Without a satisfactory theory of liquidity

premiums, predicting the effect of policies on the shape of the yield curve is almost

impossible.“.

A couple of years later Mankiw (1986) still writes: “Developing theoretically

plausible and empirically testable theories of the term premium should remain high on

the research agenda.“.

Looking back at the EH literature recalled in the present paper, it is nowadays easy

to understand its failure to fully explain the existence and the time pattern of variation

in term premia. The EH, in its various versions, is essentially a semideterministic theory

in that uncertainty is considered but not appropriately treated. Since the late seventies,

stochastic models of the TS based on the no-arbitrage equilibrium assumption have

provided a proper treatment of uncertainty.

The still growing stochastic literature on the TS can be broadly split into two main

approaches: the Arbitrage Pricing Theory Approach and the General Equilibrium

Approach. In a previous paper, Boero and Torricelli (199.5) theoretical superiority of

the latter has been assessed on the basis of the two following arguments: first, relevant

variables, such as the spot interest rate and the interest risk premium, are endogenous,

secondly the relationship between the real and the financial side of the economy . . .

becomes a clear and important element in understanding the TSvnl

The first argument is essential to the present discussion: the general equilibrium

models of the TS till a theoretical gap in that they endogenise risk premia. In fact, not

only they imply the existence of risk premia, but also explain their shape and their time

pattern of variation in relation to the characteristics of the underlying real economy.

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Within this approach, two are the papers that can shed light on the existence and the

effects of time-varying risk premia: the Cox, Ingersoll and Ross (1985b) milestone

paper and the more recent Longstaff and Schwartz (1992) paper.

Cox, Ingersoll and Ross (198513) theory of the TS is obtained by specialising Cox,

Ingersoll and Ross (1985a) general equilibrium model which gives a complete

intertemporal description of a continuous time competitive economy. The CIR model

is a single factor model having technology as the only stochastic state variable. Its main

characteristics is the endogenous nature of the stochastic process for the equilibrium

risk-free rate, which depends on the basic assumptions made about the real economy

(e.g. technology, production and the preference structure).

The CIR model not only shows the time-varying nature of the risk premium, but

offers also a precise functional form for it. The expected rate of return on a bond

derived within the CIR model is given by:

where: h is the market price for risk and depends on the characteristics of the

underlying economy, the subscript stands for partial derivative and the term in

parenthesis is the term premium at issue in this paper.

On the latter the authors state: “The factor r is the covariance of changes in the

interest rate with percentage changes in optimally invested wealth (the “market

portfolio”). Since P, < 0, positive premiums will arise if the covariance is negative.” To

see this, suppose the covariance is negative, the bond is a bad hedge because has a low

value when the marginal utility of wealth is high. Recalling a standard CAPM result, if

a financial asset is a bad hedge it has to be priced so to yield a positive premium.

Summing up, the CIR model builds up a theory of the TS which is also a theory of

time-varying term premia, still missing in the EH literature. In fact, not only their

existence is explained, but their characteristics are clarified:

i) the term premium depends on the level of the risk-free interest rate, on the

covariance with optimally invested wealth and on the interest rate elasticity of bond

prices;

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ii) its sign depends only on the covariance of changes in the interest rate with

percentage changes in optimally invested wealth.

Yet, the single factor nature of the basic CIR model implies that term premia on

different bonds are perfectly correlated. Longstaff and Schwartz (1992) overcome this

limitation by extending the CIR general equilibrium model to the case of two state

variables. The LS model is thus a two-factor model, the two factors being the short

term interest rate, r(t) as in CIR, and its volatility V(r), which is also “intuitively

appealing since volatility is a key variable in pricing contingent claims”.

The instantaneous expected return for a discount bond stemming from LS model is

the following:ix

where:

il is as before the market price for risk parameter,x

cz, fl ware parameters describing the dynamics of r(f) and V(r),

r is time to maturity and B(zj is a non-linear function of the model parameters and

of maturity 7.

As in CIR, the term premium is an increasing function of the riskless rate and its

sign is determined by the market price of risk parameter. The novelty here is that the

term premia is now a linear function of both the interest rate level and its volatility

where the relationship with the latter is indeterminate.

At this stage the question is: how is it possible to reconcile CIR and LS result on

time-varying term premia with the models presented in section l? In other words, how

can the general equilibrium theory on time-varying term premia help to explain the

inability of the EH to assess the information content of the term structure?

The answer of the present paper relies on a broader use of the word “information”

as opposed to that generally made in the literature xi In fact, in regressions (1) to (4)

the “information” in the term structure is represented only by some measure of its

slope: the information at issue in those equations is only that contained in either the

forward-spot or the long-short spread. Yet, the time-varying term premia contained in

those spreads obscure the information content of the same.

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Thus, the need of adding a time-varying term premium as an explanatory variable in

the above-mentioned regressions is apparent. Several studiesxii have already

attempted that, but none has yet done it on the basis of a proper theory of the term

premium which allows to have a theoretically consistent interpretation of the results.

Here, on the basis of the general equilibrium models recalled in this section, possible

extensions of regressions (1) to (4) are proposed.

The first is based on CIR results: from (S), the term premium to be added to the

regression equations could be modelled as the product of the market price for risk and . . .

the interest rate elasticity of bond pricesxm :

(7) @CT) = ad% t, T), r(t))

where (P(r, f, T),r(l)) is the interest rate elasticity of bond prices.

A problem with (7) is its implementation: in the CIR model depends on the

covariance between wealth and the state variable and on the preference structure and

therefore would require a previous and separate estimation. The real problem here is

that the only source of variation in the term premium is again r(t), which plays already

a role in the regressions considered in the previous section.

To move a step forward, one has to look at the implications of the LS model. From

equation (6), it is possible to write the term premium as follows:

(8) @t, T) = 1f(T - t)(ar(t) - V(t))

where f(T-t) is a non-linear function of the model parameters and of maturity.

For fixed maturity, the term premium is a linear function of both r(t) and V(t).

Unfortunately, as it stands in (S), the term premium cannot be simply added to a linear

regression equation. Yet, it suggests that the riskless rate volatility is a further source

of information which has to be incorporated into regression equations beside the

riskless rate or the yield spread. Two points still have to be discussed: how to

incorporate it and what measure of volatility to take.

As for the first, if the theory has to be testable the simplest way to preserve linearity

is to incorporate volatility by means of a second regressor in equations (1) to (4) of the

type: +rV(t) where the sign of y is, according to the LS model, indeterminate and

depends essentially on the length of the maturity. The consideration of the volatility

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term could explain some of the empirical regularities which are at odds with the EH. In

fact, even if the term structure were flat (i.e. the first regressor equal to zero), volatility

could still account for future spread or premia.

The second point is what is volatility. Theoretically, in LS paper, it is the

instantaneous variance of changes in the riskless rate and for its estimation the

GAKCH framework is suggested. It is thus clear that the point is more an empirical

one. In this paper we measure volatility by a moving average of absolute changes in the

short rate, over a predetermined number of observations. This measure has been used

in previous studies, but this choice is arbitrary and alternative proxies for a time

varying risk premium could be adopted.

Summing up, the two mentioned stochastic models of the term structure provide a

few interesting hints for future research on the information content of the term

structure. At this stage, the issue is mainly an empirical one: in fact, the opportunity to

introduce volatility (or other regressors explaining time-varying term premia) has to be

confirmed by the data. In the analysis below we present the first stages of our empirical

study of the German term structure.

4. ESTIMATION OF THE TERM STRUCTURE

The empirical results of this study are based on time series drawn from estimated

term structure curves.xi” We have estimated 134 monthly term structures of interest

rates based on German government bond data, covering the time period from November

1983 to December 1994. The data are provided by the Karlsruher Kapitalmarkt

Datenbank (KKhJDB) and refer to the 15th of each month.

The estimation precedure followed the non-linear regression approach suggested by

Chambers, Carleton and Waldman (CCW) (1984). The basic idea of CCW is to

approximate the term structure of interest rates using a polynomial of an arbitrary

degree. Formally for a fixed estimation date t the following equation holds:

(9)r(l,T)=~qT-r)'~' J=,

The discount fiutction B,(T) according to (9) is an exponential polynomial:

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(10) B,(T)= exp(-&jrj) 1’1

Based on this assumptions a statistical model to estimate the parameters of the

discount function from the observed coupon bond prices is:

(11)

n-1

8.(kS,Z)=~[~~exp(-~olll)]+(10O+~,)exp(-~~,~~)+~,. i=, j=l ,=I

where k,(s= l,...,m) denotes the coupon payment of the s th coupon bond,

r,,(i = l,...,n) denotes the time measured in years from the estimation date to the

relevant coupon payment date.

Based on (11) the parameters nj,(j=l,,..,l) can be estimated using a non-linear

regression approach.

For the given data different degrees of the polynomial were employed. In order to

assess the goodness of fit of the estimated parameters we used the mean absolute

deviation (MAD). The price deviation&, is the difference in DM of the observed market

price of bond s and its present value calculated with the estimated rates. The Mean

absolute deviation is then defined as follows:

(12) MAD = -!&bs(c,)

As mentioned above we estimated the parameters for different degrees of the

exponential polynomial. However, we eventually decided in favour of an exponential

polynomial of degree 8. Exponential polynomials of degree less then 8 turned out to

yield heavily worse MAD, whereas no significant enhancement of the goodness of fit

could be achieved when we used exponential polynomials of higher degrees.

For the performed term structure estimation we obtained an overall MAD of DM

0.190 which has to be considered an excellent result.” We have also calculated the

MAD for each year. The best fit is for 1989 with an MAD of DM 0.113 the worst fit

turns out to be for 1985 with MAD of DM 0.369. Further analysis shows that overall

75% of the absolute deviations are less than DM 0.228.wi

The regressions below use monthly observations of interest rates derived from the

estimation of the German term structure. The short rates considered are the one-day and

the one-month rates. The long rates are the one- to tive-year rates. The evolution of the

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term structure of interest rates over time is presented in Figures la and lb. Figure la

shows the time period from November 1983 to June 1989, Figure lb covers the time

period from July 1989 to December 1994. As can be better seen from the Figure 2,

German interest rates were at a relatively high level at the beginning of the time period

under consideration. German rates were heavily influenced by high rates in the U.S. due

to the so called deficit spending policy of the Reagan administration. However, interest

rates in Germany declined to a very low level in the mid-eighties The next period of

high interest rates was mainly caused by fears of inflation resulting from the monetary

union of East and West Germany and their later unification. As a result interest rates

increased sharply by the end of 1989, but were still at a moderate level (about 6%-7%).

The DM was introduced in East Germany in the middle of 1990. This resulted in tirther

inflation and caused interest rates to increase again. Especially rates for short maturities

were extremely high. After rates peaked in October 1993 a constant decrease of interest

rates could be observed.

All rates move mostly in the same direction. In particular, the short (one day rate and

one month rate) resulted almost equivalent, so that in the present paper we decided to

use only the one day rate. Fig.lshows the evolution of the one-day rate and the 5 year

rates for the period November 83 December 94.

The time the short rates are lower than the 5 years rate until 1990, implying an

upward sloping term structure, while from the end of 1990 to the begining of 1994 the

term structure was downward sloping.

5. PRELIMINARY ANALYSIS OF THE DATA

The estimated models

The following empirical study is based on regressions of the type described by

equation (3) but extended to examine the information content of the term structure for

both short and long rates. The data set consists of monthly observations of interest rates

derived from the estimation of the term structure described in the previous section.

First, we study the usefulness of the spreads between long and short rates as

indicators of future changes in both short and long rates. So we estimate the following

regressions:

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(13) rt+h - rt = a + Y @t-q) + ut+h

(14) Rt+h - Rt = a + Y @t-q) + Ut&

where r is the short rate, R the long rate and h the forecast periods (months).

Second, we add a second regressor, the volatility of the short rate, as a proxy for a

time-varying risk premium, to see whether more information can be extracted from the

term structure by including this variable. The regressions used for this second purpose

are the following:

(15) rt+h - rt = a + y (Rt-rt) + 6 VOLt + ut+h

(16) Rt+h - Rt = a + y (Rt-rt) + 6 VOLt + ut+h

As we will see later, the volatility has been approximated by a moving average of

absolute changes in the short rate, computed over six periods (MA6). This measure will

clearly be a crude proxy of the true volatility, but it will be related to it, and it has been

used in other studies (see Simon, 1989).

Most series exhibit a structural break during the German Unification period.

However, the empirical analysis below is not affected by this regime shift as much as we

would expect, perhaps because the breaks cancell across linear combinations of the

variables (see the concept of co-breaking series recently introduced by Hendry and

Clements (1995)). This point requires further examination.

Unit root tests

Estimation of regressions of the type described by equations (13) - (16) requires that

the variables are stationary. So we first checked for the presence of a unit root by

applying Augmented Dickey-Fuller (ADF) tests. The ADF test accounts for temporally

dependent and heterogeneously distributed errors by including lagged innovations

sequences in the fitted regression.

The results of the tests not reported here (see Boero-Madjlessi.Torricelli, 1995),

suggest that changes in the short interest rate are stationary for forecast horizons of one-

to 16-months, but they indicate the presence of a unit root for longer horizons. Changes

in the long rate are stationary for forecast horizons of one- to nine-months and become

non-stationary for longer horizons. Finally, the tests suggests that all spreads and the

811

volatility term are stationary. These results are important for determining which

variables should be considered in the regression analysis performed below.

6. TESTS OF THE INFORMATION IN THE GERMAN TERM STRUCTURE

The results presented below are derived from estimates of equations (13) and (14).

First we address the question whether current spreads contain information for future

changes in the short rate. Then we address the same issue for changes in the long rate.

6.1 Predicting short term rates

The forecasting equations relate used for this analysis relate changes in the short term

interest rate h periods (months) ahead to the current spread between long term

(R=1,2,3,and 5 years) and short term (t=l-day) interest rates. The regressions are

similar to those used in Fama( 1990). To obtain meaningful results, all the variables in a

regression should be stationary or cointegrated. According to the unit root tests in the

previous section, all spreads are stationary, while changes in the short rate are stationary

for forecast periods up to 16 months, and become non stationary over longer periods.

So the estimated equations in Table 1 are specified for changes in the short rates from 3-

to 12-months ahead, with the spreads of the yield on a one, two, three and five-year

bond over the one-day spot rate. These are called in table 1 SPl-0, SP2-0, SP3-0 and

SP5-0, respectively. We do not report results with SP4-0 as they were very similar to

those with SPS-0. Due to overlapping data, the equations are estimated by OLS with

corrections based on Newey-West (1987) for a moving average of order h-l, where h is

the forecasting horizon (in months), and for conditional heteroscedasticity. Several

findings are of interest.

First, our results indicate that the German term structure, in the period 1983-1994,

appears to forecast future interest rates surprisingly well. The results reveal that the

parameter estimates for 7, the coefficient of the slope of the term structure always

contain significant predictive power for future changes in short-term rates at the 1

percent level, with R* ranging from 0.15 to 0.48. We note that these R2s are much

higher than those obtained with data for the US term structure. For example, Fama

(1990) in table 1 reports R*=O.OO for interest rate changes one-year ahead, and R*=O.24

for forecast horizon extended to 5 years. This finding is in line with recent evidence in

812

Hardouvelis (1994), where using different data (not from the estimation of the term

structure) and methodology notes that the US term structure seems to predict interest

rates less well than the term structure for the other G-7 countries.

Second, Table 1 shows that, for a common spread, the coefficient estimate y and the

predictive power of the regressions increase with the forecast horizon. For example,

when the one-year spread (SPI-0) is used as the regressor, for short forecast horizons (3

to 6 months) the coefficient estimates are less than one, and become greater than one as

the forecast horizon is extended (10 to 12 months). Similarly, the R2 values increase

from 0.27 (3-months ahead) to 0.47 (l2-months ahead). The finding that the forecast

power increases with the forecast horizon is in line with the evidence for the US in Fama

(1984, 1990), Fama and Bliss (1987), Campbell and Shiller (1987), Hardouvelis (1988)

and Mishkin (1988). These studies, however, report much lower R% than those we

obtained for Germany.

Third, our results suggest that the one-year spread contains more information for

changes in the short rate over periods closer to one year; in the same way and in

accordance with equation (3), we would expect a 2-year spread to be more informative

for changes in the short rate over a period of two years. However, we could not

investigate this issue further, using this simple regression framework, due to the non

stationary properties of changes in the short rate over periods longer than one year.

These results are interesting as they indicate that the choice of the spread to include in

the regressions matters. This is in contrast with previous findings for the US (see Fama

1990), where a common spread (five-year spread) could be used in all regressions as

different spreads were very highly correlated and shared nearly identical time-series

properties. Our results suggest a common pattern within each forecast horizon

considered: the most informative spread is the one-year over one-day rate (SPl-0), and

the values of both y and R2 decrease with the length of the maturity spread

To summarize this section, overall, the results for the German term structure are

surprisingly good, compared with previous findings for US data. In most cases the

“slope” of the term structure has a fair degree of explanatory power for the change in

short rates, when the appropriate spread is used, with R2 of 0.27 and above. These

results are even more satisfactory, given the simplicity of these regressions: so far we are

813

restricting ourselves to extract information about future changes of short interest rates

using only knowledge at time t of the “slope” of the term structure. Surely the spread is

not the only predictor for changes in the short rate, Highly significant diagnostic tests

can be considered evidence for omitted variables, Moreover, a plot of recursive

estimates of the y coefficient in Figure 10 reveals some instability and suggests that the

equations should be better specified. In section 7 we make use of an additional variable,

specifically the volatility of the short term interest rate, to see if we can extract more

information from the term structure beyond that contained in the slope. Before we do

that we repeat the analysis above for long term rates,

6.2 Predicting long-term interest rates

An extensive literature, mostly on US data, documents that the term structure is a

good predictor for variations in the short-term interest rate, but not for the long term

(see Mankiw, 1986, Campbell and Shiller, 1991, Hardouvelis, 1994, and Grande, 1994,

for an application to the Italian Treasury Bill market). A general finding of these studies

is that if the slope of the term structure forecasts changes in the long rate at all, the sign

of the coefficient estimate is negative, that is the spread predicts changes in the long rate

in the direction opposite to that predicted by the Expectations Hypothesis. In this

section we test the information content of the German term structure by regressing

changes in the long-term interest rate on the slope of the term structure. Table 2 reports

the results. The long term rate Rt is the five-year rate, the short rate rt is the one-day

rate. The analysis is conducted for l- to 9-months forecast periods only, as according to

the unit root tests, changes of the long-term interest rate over periods longer than 9

months show non-stationarity. Several findings are of interest.

First, in line with previous studies where different data and methodologies have been

used, in these regressions which have the change in the long rate as the dependent

variable, the results are overall negative, suggesting that there is no predictive power in

the slope of the term structure for long-rate movements. The coefficient estimates of the

spread are not significantly different from zero, and the R2 are very low.

Second, although the predictive power of the regressions is poor, it is interesting to

note that while previous studies with data for the US report a negative sign for the

coefficient y, indicating that the spread predicts the wrong direction in future changes of

814

the long rate, the direction emerged in our estimates for Germany is correct, according

to the expectations hypothesis. However, due to the poor specification of the model, we

cannot interpret these results further.

7. THE INFORMATION CONTENT OF VOLATILITY

In this section we extend the framework presented in section 6, used in standard tests

of the Expectations Hypothesis, by including a measure of the riskless rate volatility in

the information set, as a proxy for a time-varying risk premium. The results presented

below are derived from estimates of equations (15) and (16), which are inspired by the

Longstaff and Schwartz (1992) model, as discussed in section 3. As this analysis is quite

new, it can only be viewed as an early experiment.

First, we address the issue whether a volatility term, proxy for a time-varying risk

premium, can help extracting more information from the term structure for future

changes in the short rate. Then, we see whether inclusion of a time-varying risk premium

can improve the forecast performance of the regressions for the long rate.

7.1 Does volatility add information for the short-term interest rate?

In table 3 we extend the standard framework presented in section 6 by including a

measure of the riskless rate volatility in the information set, as a proxy for a time-varying

risk premium. The risk premium is proxied by a moving average of absolute one-day

interest rate changes over the previous six months. A similar measure has been used in

previous empirical analyses of the term structure (see Simon, 1989, for example), but of

course the choice is arbitrary, and comparisons with other measures should be explored

in future research. A number of results are of interest.

First, the coefficient estimates of y remain significant at the 1 per cent level in all

cases.

Second, the coefficient estimate 6 of the volatility term is significantly different from

zero in ten cases out of sixteen. Overall, however the contribution of the short rate

volatility in terms of R2 is marginal. Thus, most of the information in the term structure

for movements in the short rate is contained in the spread.

Third, despite the marginal increase in the predictive power, diagnostic tests suggest

that these extended regressions represent an improvement in some cases. Moreover,

815

recursive estimates of the showed a remarkable improvement in terms of stability in the

extended regressions. Indeed, both coefficients y and 6 are surprisingly stable, and seem

not to be affected by the structural break observed in most series around the German

Unification period. A more thorough analysis of the impact of regime shifts in the

estimated models is currently under investigation.

Finally a note on the negative sign of the volatility term. Although this sign is in line

with previous studies which have used similar proxies (see Simon, 1989), according to

the theoretical model of Longstaff and Schwartz (1992) the sign of volatility is

indeterminate. So, as our approach is inspired by the LS model, we are inclined to

believe that the finding of a negative sign is specific to the particular operational

measure used for the short rate volatility, and to the data and period considered. As a

preliminary check, not reported here, we conducted the analysis for different subperiods,

and with the standard deviation of the short rate as an alternative measure for volatility.

The results were not very robust in terms of both the sign and the significance of the

estimated coefficient of the volatility term. These depend on the forecast periods

considered, the set of interest rates and spreads selected, and the proxy used for the

volatility term. The choice of the volatility measures is in fact entirely arbitrary, and it is

not clear what is the best measure to use. Comparisons with other indices should be

further explored.

7.2 Does volatility add information for the long-term interest rate?

In table 4 we extend the regressions presented in section 6.2 for the long-term

interest rate, by including a time-varying risk premium term. To facilitate the

comparison, we also report the results from table 2. The proxy used for volatility is the

same as in section 7.1, that is a moving average of absolute one-day interest rate

changes over the previous six months, The results presented below show that, with the

inclusion of the volatility term, the forecasting regressions for changes in the long rate

significantly improve. In particular, the coefficient estimate y now becomes significant at

the 5 and 1 percent levels with a coefficient which is more than twice the value obtained

without the volatility term. The coefficient ‘estimate of 6 is also significantly different

from zero, and there is a fair improvement in terms of the R2

816

Moreover, a recursive estimate of the coefficient has revealed that, with the inclusion

of the volatility term, the coefficient estimate of y becomes stable, and that the value of

the coefficient is positive for most of the sample period. Thus, our results suggest that

the direction indicated by the spread for future changes in the long rate is correct,

according to the Expectations Hypothesis, in contrast with previous studies with U.S.

data.

To conclude, our analysis suggests that these extended regressions represent an

improvement with respect to simple regressions which use only knowledge at time t of

the slope of the term structure. However, there is still some information missed about

future changes of interest rates. In particular, dynamics are excluded from these

regressions, and ways to improve the specification of these models are under study. One

possibility is to adopt an error correction specification which combines the long-run

information in the term structure with short-run dynamics. Another possibility is to use

the conditional variances estimated from GARCH models, rather than a moving average

term as a proxy for interest rate volatility.

8. CONCLUSIONS

Previous studies have recognised the failure of the Expectations Hypothesis to

properly detect the information content of the term structure. In the present paper, we

have proposed an explanation for such a failure based on the existence of time-varying

term premia and we have tested it for the case of Germany.

The issue is reexamined in the light of the most recent stochastic models, which

essentially confirm the existence of time-varying term premia. In order to combine the

latter with the information analysis, an explicit theory for the time-varying term premia is

needed. Such a theory is found only within general equilibrium stochastic models of the

term structure. To this end, Cox, Ingersoll and Ross (1985) and Longstaff and Schwartz

(1992) models are examined and their implications about term premia considered.

Among other things, term premia turn out to be proportional not only to the level of the

riskless rate, but also to its volatility.

Hence, the theory suggests the need for a broader use of the expression “information

in the term structure” which, in our opinion, implies consideration of a second regressor

817

in equations (1) to (4). According to the general equilibrium theory of the term

premium, a sensible explanatory variable is volatility.

The empirical analysis performed here is quite preliminary and can be viewed as an

early experiment. However, several results are of interest. First, our results indicate that

the German term structure appears to forecast future short term interest rates

surprisingly well, compared with previous studies with US data.“” Second, and in line

with previous findings, the slope of the term structure alone does not have predictive

power for long term interest rate movements, though the direction suggested by the

coefficient estimates is correct. Third, inclusion of a volatility term represents an

improvement in most regressions for the short rate, and significantly improves those for

the long rate. The empirical results improve in terms of the statistical significance of the

coefficients, their stability, and the predictive power of the regressions.

Although these results seem overall interesting, the regressions used for the test of

the information content of the term structure are very simple, and so there are many

directions this study can take to improve the predictive power of the models and better

understand the nature of the series.

In particular dynamics are excluded from the analysis. One way to improve the

specification of the models is to adopt an error correction (or equilibrium correction)

specification which combines the long-run information in the term structure with short-

run dynamics. Moreover, further research is needed to establish whether a more

significant contribution to the forecasting power of the term structure can be obtained

with alternative measures of volatility. One possibility is to use the conditional

variances estimated from GARCH models, Finally, the series used in this study contain

a regime shift due to the German unification; once account is taken of the structural

break, the results may improve significantly. The points just made represent current

and future research work.

APPENDIX: Notation and definitions

The literature on the term structure of interest rates is unfortunately characterised by

a cumbersome notation, which is furthermore far from being uniform among

contributors. In the present appendix Shiller( 1990) is adopted.

818

Let:

. f be time with t s T

p(t,T) be the price at t of a pure discount bond maturing at T, whose

principal is 1

. m = T - t be the term or time to maturity of the bond

then r(t,T), the yield to maturity or spot interest rate of the pure discount bond

maturing at T, whose principal is 1, is given by:

r(t,T)=-In(p(t,T))/(T-t).

r(t, T) is the short rate if the term m is small, the long rate if m is big.

The forward rate at time t applying to the time interval t ’ to T is given by:

f(t,t’,T)=-In(p(t,T)/p(t,t’))/(T-t’) with t<t’<T

or alternatively:

f(t t, ,)=(T-t)r(t,T)-(T-t’)r(t,t’) , , T-t’

The holding period return from t to t’ on a bond maturing at T is given by:

h( t, t’ , T) = (T- t)r(t,T) -(T- t’)r(t’,T)

t-t

Having defined relevant returns, it is now possible to define the term premia that play

a role in the analysis of the informative content of the term structure.

The forward term premium is defined as the difference between the forward rate and

the corresponding future spot rate, i.e.:

@r(t,t’,T)=f(t,t’,T)-Enr(t’,T) with t<t’<T

and Ei is the rational expectation operator.

The holding period term premium is defined as the difference between conditional

expected holding period yield and the corresponding spot rate, i.e.:

&(t,t’,T) =Eth(t,t’,T)-r(t,t’) with t<t’<T

The rollover term premium is defined as the difference between bond maturing at

time t ’ and conditional expected holding period return from rolling over m-period bonds,

i.e.:

@,(t,V,m) = r(t,t’)-Eth(t,t’,t+m) with t<t+m<t’

819

In the early literature, e.g. Hicks (1946) and Keynes (1936), the forward and the

rolling over premia were referred to as “risk premium” or “liquidity premium”. The

holding period term premium is the theory of finance “excess return”.

See Shiller (1990), for the relationships between the various types of premia.

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Cox J., Ingersoll J.E. and S.A.Ross, 1985a, An intertemporal general equilibrium model of asset pricing, Econometrica, 53, 363-384.

Cox J., Ingersoll J.E. and S.A.Ross, 1985b, A theory of the term structure of interest rates, Econometrica, 53, 385-407.

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Fama E.J., 1976a, Forward rates as predictors of future spot rates, Journal of Financial Economics, 3, 3 6 l-3 77.

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Fama E.J., 1990, Term-structure forecasts of interest rates, inflation, and real returns, Journal of Monetary Economics, 25, 59-76.

Fama E.J. and R.R. Bliss, 1987, The information in long-maturity forward rates, American Economic Review, 77, 680-692.

Engle, R.F., D.M.Lilien and R.P.Robins, 1987, Estimating time-varyuing risk- premia in the term structure: The ARCH-M model, Econometrica, 55, 391-407.

Estrella A. and G.A. Hardouvelis, 1991, The term structure as a predictor of real economic activity, Journal of Finance, 46. 555-576.

Evans M.D.D. and K.K. Lewis, 1994, Do stationary risk premia explain it all? - Evidence from the term structure, Journal of Monetary Economics, 33,285-3 18.

Gerlach S., 1995, The information content of the term structure: evidence for Germany, Bank of International Settlements, W.P. n.29.

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Grande, G., 1994, La curva dei rendimenti dei BOT come misura dei tassi futuri attesi, Bank of Italy, discussion paper No. 232.

Hardouvelis G.A., 1994, The term structure spread and future changes in long and short rates in the G7 countries - Is there a puzzle?, Journal of Monetary Economics, 33, 255-283.

Hendry D.F. and M.P. Clements, 1995, The Econometrics of Macro-economic Forecasting, paper presented at the European Economic Association Conference, Prague, September 1-4.

Jorion P. and F. Mishkin, 1991, A multicountry comparison of term-structure forecasts at long horizons, Journal ofFinancial Economics, 29, 59-80.

Lauterbach L., 1989, Consumption volatility, production volatility, spot-rate volatility and the returns on treasury bill and bonds, Jownal of Financial Economics, 24, 155-179.

Longstasff F.A. and E.S.Schwartz, 1992, Interest rate volatility and the term structure: a two-factor model, JozrrnalofFinance, 47, 1259-1282.

Mankiw N.G., 1986, The term structure of interest rates rivisited, Brookings Papers on Economic Activity, 1, 6 l-96.

Mankiw N.G. and L.H. Summers, 1984, Do long-term interest rates overreact to short-term interest rates?, Brookirgs Papers on Economic Activity, 1, 61-96

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Mishkin F., 1990, What does the term structure tell us about future inflation?, Journal of Monetary Economics, 25, 77-95.

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Simon, D.P., 1989, Expectations and risk in the Tresury bill market: An instrumental variables approach, Journal of Financial and Quantitative Analysis, 24, 357-365.

Torricelli C., 1995, The information in the term structure of interest rates. Can stochastic models help in resolving the puzzle?, Materiali di Discussione, Universitri di Modena, n 113.

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(‘) The authors wish to thahk Ignazio Angeloni and Giuseppe Marotta for helpful comments on an earlier version of the present paper and Hermann Gdppl for providing research support. G. Boero aknowledges financial support from CRENoS and Progetto Vigoni (CRUI). C. Torricelli acknowledges financial support from MURST 40% 93-94 and CNR 94.00087.CTOl and Progetto Vigoni (CRljI).

’ An earlier version of this paper can be found in Boero, Madjlessi and Torricelli (1995).

ii For a survey on the literature, see: Boero and Torricelli (1994, 1995), Vetzal(l994). “’ The information we address in this paper is only information on monetary variables (Interest rates, inflation) and not on real variables. Yet, the term structure can be used as an indicator and predictor of real economic activity: among others see Estrella and Hardouvelis (1991).

821

iv For a discussion of various versions of the EH in relation to the information issue, see Torriceili (1995). ” The empirical literature considered in this section uses data drawn from estimated term structure curves which can be considered as returns on pure discount bonds. In the next sections, some reference is made also to papers based on yield data. By yield data we mean the term structure of the internal rate of return on coupon bonds, which is not a correct measure of the term structure of interest rates. yi In two previous papers, Fama (1976a,b), the author shows that the forward rate f(t,T-1,T) can be decomposed in three parts: (i) the expected value of the future spot rate, E[r(T-l,T)], (ii) the expected holding premium in the one month return on a bond of maturity T, (iii) current and expected changes in future expected holding premia. Precisely the latter two components obscure the relationship between f(t,T-1,T) and E[r(T-l,T)] vii For an illustration of this finding, see Fama and Bliss (1897) page 690. yiii For a complete comparative evaluation of the two approaches, both at a theoretical and at an empirical level, the interested reader is referred to the original article. ix Details about the model and the variables involved in (6) are to be found in the original paper. Here only results concerning the term premium are discussed. x In LS model as well can be interpreted in terms of covariance between optimally invested wealth and the state variable: see Longstaffand Schwartz (1992) page 1263. xi For example, Mishkin (1990) writes: “...we are restricting ourselves to extracting information about the future path of inflation using only our knowledge of the slope of the term structure.” xii Among others: Shiller, Campbell and Schoenholtz (1983) Engle, Lilien and Robins (1987), Simon (1989). xiii Since the regression equations proposed in section 2 are in discrete time, also the term premia proposed in this section should be added in their discrete time version. xiv Other studies use Bundesbank bond yield data (e.g. Gerlach, 1995). The Bundesbank provides the term stucture of internal rate of returns of default-free bonds, which coincides with the term structure of interest rates only if the latter is flat. Since the aim of the present paper is to detect the informative content of the term structure of interest rates, we think it is important to use time series drawn from properly estimated term structure curves. xv Biihler, Uhrig, Walter and Weber (1995) in a recent study tit a cubic spline to a discrete term structure estimated with a quadratic programming approach. They report an MAD for 1993 of DM 0.114. wi Further details on the estimation of the term structure are available upon request to the interested reader. ‘“Ii A possible intuitive interpretation of such a difference can be the following. German term premia are relatively small and constant, compared with U.S., so that most of the information about future short rates is given by the yield spread. This would also explain why volatility does not add much information.

822

Table 1 Estimated regreSSiOfl: rt+h- rt = a f y (Rt-rt) + ut+j,

12 SPl-0 1.41*** 0.47 *** 0.21 0.58 ** 84.11-94.12 12 SP2-0 1.27~~~ 0.46 *** * * *** 122 12 SP3-0 l.lo*** 0.42 *** * 0.16 *** 122 12 SP5-0 o.g4*** 0.39 *** 0.13 0.29 *** 122

***, **, and * indicate statistical significance at the I%, ** 5% and * 10% levels, respectively. LM(p) is the Lagrange multiplier test for pth-order serial correlation FF( 1) is the Ramsey-RESET test for functional form misspecification N(2) is the Bera-Jarque test for normality H(1) is the White test for heteroscedasticity

823

Table 2 Estimated regression: Rt+h - Rt = a + y (Rt-rt) f ut+h

R = j-year interest rate r= one-day interest rate

Table 3 Estimated rcgrcssion rt+l, - rt = a + y (R,-rt) + 6 VOLt + ut+l,

10 ISPI- I 13x***l-0 19 104x I*** In37 IO81 I** 1X53-9417 IO 10 10

_._. &-tj i:;;*** -I 7-t* lnd7 I***

I _.__ __._ _ .._- ..e< “,.,

Ii’d; In 16 *** V. ._I Y.. 118

SP3-0 1.22*** -L.-- 2 13*** lo47 I*** 1054 101: , . ..2 *** 118 SPS-0 l.Ol*** -2.78*** IO.43 1 *** IO.53 IO.25 *** 118

I 12 SPI-0 1.37*** 0.73 0.46 *** * 0.53 ** 85.5-94.12 12 SP2-0 1.34*** -0.34 0.44 *** 0.14 * ** 116 12 SP3-0 1.27*** -1.23 0.42 *** 0.16 ** ** 116 12 SPS-0 1.04*** -1.91* 0.39 *** 0.19 0.14 *** 116

***, **, and * indicate statistical significance at the l%, 5% and 10% levels, respectively. LM(p) is the Lagrange multiplier test for pth-order serial correlation; FF( 1) is the Ramsey- RESET test for functional form misspecifxation; N(2) is the Bera-Jarque test for normality; H(l) is the White test for heteroscednsticity.

824

Table 4

Estimated regression: Rt+h- Rt = c( f y (Rt-rt) + 6 VOL, + q+h

R = 5-year interest rate r = one-day interest rate

9 1 SPS-0 [ 0.21 1 0.07 1 84.8-94.12 1 125 I 10.43** j-1.73* [ 0.17 ) 85.3-94.12 1 11s

*** and ** indicate statistical significance at the 1% and 5% levels, respectively. VOL= volatility of the short term interest rate proxied by a moving average of absolute one-period changes over the previous six months

825

-----4

Fipre la, ~1,~ term stnlcture of German interest rates: 111gs3 - 6’198g

FigLlre lbL The tern, gructLire of German interest rates: 1D989 - 6’1g9

co-

--L